Computable error bounds for approximate periodic solutions of autonomous delay differential equations

In this paper, we prove a result that says: Given an approximate solution and frequency to a periodic solution of an autonomous delay differential equation that satisfies a certain noncriticality condition, there is an exact periodic solution and frequency in a neighborhood of the approximate solution and frequency and, furthermore, numerical estimates of the size of the neighborhood are computed. Methods are outlined for estimating the parameters required to compute the errors. An application to a Van der Pol oscillator with delay in the nonlinear terms is given.     

Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains

The exact solutions for the motion of a Maxwell fluid due to longitudinal and torsional oscillations of an infinite circular cylinder are determined by means of the Laplace transform. These solutions are presented as sum of the steady-state and transient solutions and describe the motion of the fluid for some time after its initiation. After that time, when the transients disappear, the motion is described by the steady-state solution which is periodic in time and independent of the initial conditions. Finally, by means of graphical illustrations, the required times to reach the steady-state are determined for sine, cosine and combined oscillations of the boundary.     

Exact analytical solutions for forced undamped Duffing oscillator

The steady-state response of an undamped Duffing oscillator to periodic external forces is studied. The forcing functions are chosen such that the time course of the displacement can be described by exact analytical expressions. The displacement and forcing functions are governed by Jacobi elliptic functions and thus are periodic but generally non-harmonic. The parameter of the elliptic functions is deliberately chosen. For certain parameter choices, exact analytical expressions are found for the frequency–amplitude relation.     

1/3 subharmonic solution of elliptical sandwich plates

IntroductionInrecentyears,withtheessentialadvantageoflightweightandhighrigidity ,sandwichplatesandshellshavebeenusedasanimportantpatternofstructuralelementsinaeronautical,astronauticalandnavalengineering .However,nonlinearproblemsforsandwichplatesandshellsareonlyinvestigatedbyafewbecauseofthedifficultiesofnonlinearmathematicalproblems.LiuRen_huaiandXuJia_chu[1,2 ]andothershavemadesomeinvestigationsinthisfield .Bifurcationofnonlinearvibrationforsandwichplateshasnotyetbeeninvestigated .Inthisp…     

Integrable stationary solution for the fully nonlinear local induction equation describing the motion of a vortex filament

In this brief communication, we demonstrate an implicit exact stationary solution to the fully nonlinear local induction equation describing the motion of a vortex filament. The solution, which is periodic in the spatial variable, is governed by a second-order nonlinear equation that has two exact first integrals.     

Quasi-Periodic Oscillations,Chaos and Suppression of Chaos in a Nonlinear Oscillator Driven by Parametric and External Excitations

An analysis is given of the dynamic of a one-degree-of-freedom oscillator with quadratic and cubic nonlinearities subjected to parametric and external excitations having incommensurate frequencies. A new method is given for constructing an asymptotic expansion of the quasi-periodic solutions. The generalized averaging method is first applied to reduce the original quasi-periodically driven system to a periodically driven one. This method can be viewed as an adaptation to quasi-periodic systems of the technique developed by Bogolioubov and Mitropolsky for periodically driven ones. To approximate the periodic solutions of the reduced periodically driven system, corresponding to the quasi-periodic solution of the original one, multiple-scale perturbation is applied in a second step. These periodic solutions are obtained by determining the steady-state response of the resulting autonomous amplitude-phase differential system. To study the onset of the chaotic dynamic of the original system, the Melnikov method is applied to the reduced periodically driven one. We also investigate the possibility of achieving a suitable system for the control of chaos by introducing a third harmonic parametric component into the cubic term of the system.     

Bifurcation Analysis of Parametrically Excited Rayleigh–Liénard Oscillators

A parametrically excited Rayleigh–Liénard oscillator is investigatedby an asymptotic perturbation method based on Fourier expansion and timerescaling. Two coupled equations for the amplitude and the phase ofsolutions are derived and the stability of steady-state periodic solutionsas well as parametric excitation-response and frequency-response curvesare determined. Comparison with the parametrically excited Liénardoscillator is performed and analytic approximate solutions are checkedusing numerical integration. Dulac's criterion, thePoincaré–Bendixson theorem, and energy considerations are used in order to study the existence and characteristics of limit cycles of the twocoupled equations. A limit cycle corresponds to a modulated motion forthe Rayleigh–Liénard oscillator. Modulated motion can be also obtainedfor very low values of the parametric excitation, and in this case, anapproximate analytic solution is easily constructed. If the parametricexcitation is increased, an infinite-period bifurcation is observed because the modulation period lengthens and becomes infinite, while themodulation amplitude remains finite and suddenly the attractor settlesdown into a periodic motion. Floquet's theory is used to evaluatethe stability of the periodic solutions, and in certain cases,symmetry-breaking bifurcations are predicted. Numerical simulationsconfirm this scenario and detect chaos and unbounded motions in theinstability regions of the periodic solutions.     

Non-linear oscillator limit cycle analysis using a time transformation approach

A method is presented for the analysis of limit cycle behavior of autonomous non-linear oscillators characterized by second order ordinary differential equations containing a small parameter. The method differs from the classical perturbation methods in that the dependent variable is not expanded in a power series in the small parameter. Rather, a new independent variable is sought such that in its domain the motion is simple harmonic. Use of this time transformation technique to generate limit cycle phase portrait, amplitude and period is presented. We show results of the application of the method to the van der Pol oscillator, to an oscillator with quadratic damping, and to a modified van der Pol oscillator which is statically unstable in the limit of small motion.     

Response of a non-linearly damped oscillator to combined periodic parametric and random external excitation

An analytical solution, using the Fokker-Planck-Kolmogorov equation, is obtained for the problem of response of a non-linearly damped oscillator to combined periodic parametric and random external excitation. The solution yields first-order probability densities of amplitude and phase. These expressions are employed to distinguish between oscillations excited by external and parametric periodic forces in the presence of additional broadband random external excitation. Through decoupling of fast and slow motions an approximate expression is obtained for expected value of time to phase “switch”.     

Perturbation methods for nonlinear autonomous discrete-time dynamical systems

Two perturbation methods for nonlinear autonomous discrete-time dynamical systems are presented. They generalize the classical Lindstedt-Poincaré and multiple scale perturbation methods that are valid for continuous-time systems. The Lindstedt-Poincaré method allows determination of the periodic or almost-periodic orbits of the nonlinear system (limit cycles), while the multiple scale method also permits analysis of the transient state and the stability of the limit cycles. An application to the discrete Van der Pol equation is also presented, for which the asymptotic solution is shown to be in excellent agreement with the exact (numerical) solution. It is demonstrated that, when the sampling step tends to zero the asymptotic transient and steady-state discrete-time solutions correctly tend to the asymptotic continuous-time solutions.     

Dynamics of a balanced rotor under the action of an elastic force with a hysteresis characteristic

Recently, the constructions of rotors rotating on passive-type magnetic bearings designed with the use of high-temperature superconductors have been developed [1, 2]. One still open problem in the stability analysis of rotors on such bearings is the problem on the influence of the bearing rigidity characteristic on the system dynamics. The stability of a steady-state motion of a rotor with arbitrary eccentricity is studied in [3]. The present paper deals with the dynamics of a steady-state motion of a balanced rotor rotated by an infinite-power motor; in this case, the elastic force acting on the rotor has a hysteresis characteristic [4]. The equations of motion are described by a nonautonomous system of differential equations which is reduced with prescribed accuracy to an autonomous system, and the latter is then analyzed [5].     

Steady vortex flows of a self-gravitating gas

An exact solution of equations of steady motion of a self-gravitating gas is found. This solution describes a vortex flow of the gas from the surface of a spherical source and is partly invariant with respect to the group of rotations (Ovsyannikov vortex, singular vortex). The factor-system of the solution is reduced to finite formulas and one ordinary differential equation of the third order. Various regimes of gas motion described by this solution are determined: unlimited spreading of the gas with swirling from the surface of a spherical source and gas exhaustion with formation of a sphere with elevated density at a finite distance from this source.     

Finite amplitude,periodic motion of a body supported by arbitrary isotropic,elastic shear mountings

The undamped, finite amplitude, periodic motion of a load supported symmetrically by arbitrary isotropic, elastic shear mountings is investigated. Conditions on the shear response function sufficient to guarantee periodic motions for finite shearing with arbitrary initial data are provided. Some general results applicable for all simple shearing oscillators in the class are derived and illustrated graphically. The mechanical response of the general nonlinear shearing oscillator is compared with the response of a certain linear oscillator of comparable design. As consequence, certain static and dynamic aspects of the motion of an arbitrary nonlinear oscillator supported by shear springs are compared with those of a simple, linear oscillator for which the response is well-known and readily determined for the same initial data. The effect of a finite static shear deformation on the frequency equation for superimposed, small amplitude vibrations of the load is examined. The general analysis is applied to a class of hyperelastic biological tissues; and the frequency relation for finite amplitude oscillations of a load supported by soft tissue is derived. The finite amplitude oscillatory shearing of a general isotropic elastic continuum is described; and three universal relations connecting the stress and the oscillatory shearing deformation for every isotropic elastic material are presented.     

粘弹性桩的混沌运动分析

研究了在轴向载荷和周期性横向载荷共同作用下非线性粘弹性嵌岩桩的混沌运动情况。假定桩和土体分别满足Leaderman非线性粘弹性和线性粘弹性本构关系，得到的运动方程为非线性偏微分．积分方程；利用Galerkin方法将方程简化为非线性常微分一积分方程，同时利用非线性动力系统中的数值方法，进行了数值计算，得到了不同载荷参数、几何参数、材料参数时粘弹性桩发生周期运动、多周期运动及混沌运动的时程曲线、相图、功率谱、Poincare截面图，同时得到了挠度-载荷、挠度-几何参数、挠度-材料参数等分叉图，考察了各种参数的影响。数值结果表明非线性粘弹性桩在一定的条件下可以通过倍周期分叉的方式进入混沌运动状态，且桩的载荷参数、几何参数、材料参数对其运动状态有较大的影响。     

Averaged description of a fluid containing gas bubbles

An averaged equation is derived that describes the irrotational flow of an ideal incompressible fluid due to the expansion and translational motion of bubbles at the sites of a periodic lattice. The calculation of the coefficients of the averaged equation reduces to the solution of a cell problem. An exact solution to the problem is constructed in the form of a series in periodic harmonic functions. An infinite system of equations is written down for the coefficients of the series, and the system is analyzed asymptotically at low volume concentrations c of the bubbles.     

Solar sail dynamics in the three-body problem: Homoclinic paths of points and orbits

In this paper we consider the orbital dynamics of a solar sail in the Earth-Sun circular restricted three-body problem. The equations of motion of the sail are given by a set of non-linear autonomous ordinary differential equations, which are non-conservative due to the non-central nature of the force on the sail. We consider first the equilibria and linearisation of the system, then examine the non-linear system paying particular attention to its periodic solutions and invariant manifolds. Interestingly, we find there are equilibria admitting homoclinic paths where the stable and unstable invariant manifolds are identical. What is more, we find that periodic orbits about these equilibria also admit homoclinic paths; in fact the entire unstable invariant manifold winds off the periodic orbit, only to wind back onto it in the future. This unexpected result shows that periodic orbits may inherit the homoclinic nature of the point about which they are described.     

Averaging method for the solution of non-linear differential equations with periodic non-harmonic solutions

While Krylov and Bogolyubov used harmonic functions in their averaging method for the approximate solution of weakly non-linear differential equations with oscillatory solution, we apply a similar averaging technique using Jacobi elliptic functions. These functions are also periodic and are exact solutions of strongly non-linear differential equations. The method is used to solve non-linear differential equations with linear and non-linear small dissipative terms and/or with time dependent parameters. It is also shown that quite general dissipative terms can be transformed into time-dependent parameters. As a special example, the Langevin (collisional) equation of motion of electrons in a neutralizing ion background under the influence of a time and space-dependent electric field is presented. The method may also be used for non-linear control theory, dynamic and parametric stabilization of non-linear oscillations in plasma physics, etc.     

强非线性动力系统的频率增量法

提出一类强非线性动力系统的暧时频率增量法,将描述动力系统的二阶常微分方程,化为以相位为自变量、瞬廛频率为未知函数的积分方程;用谐波平衡原理,将求解瞬时频率的积分问题,归结为求解以频率增量的Fourier系数为独立变量的线性代数方程组;给出了若干例子。     

Delay equations with fluctuating delay related to the regenerative chatter

The mathematical models representing machine tool chatter dynamics have been cast as differential equations with delay. In this paper, non-linear delay differential equations with periodic delays which model the machine tool chatter with continuously modulated spindle speed are studied. The explicit time-dependent delay terms, due to spindle speed modulation, are replaced by state-dependent delay terms by augmenting the original equations. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. The reduced bifurcation equation is obtained by making use of Lyapunov-Schmidt Reduction method. By using the reduced bifurcation equations, the periodic solutions are determined to analyze the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions near the new stability boundary.     

Simultaneous combination and 1:3:5 internal resonances in a parametrically excited beam-mass system

The nonlinear dynamics of a base-excited slender beam carrying a lumped mass subjected to simultaneous combination parametric resonance of sum and difference type along with 1:3:5 internal resonances is investigated. Method of normal form is applied to the governing nonlinear temporal differential equation of motion to obtain a set of first-order differential equations which are used to obtain the steady-state, periodic, quasi-periodic and chaotic responses for different control parameters viz., amplitude and frequency of external excitation and damping. Frequency response, phase portraits, time spectra and bifurcation diagram are plotted to visualize the system behaviour with variation in the control parameters. Here, two distinct zones of trivial instability, blue sky catastrophe phenomena, jump down phenomena, simultaneous occurrence of periodic and chaotic orbits, period doubling of the mixed-mode periodic orbits leading to chaos, attractor merging crisis, boundary crisis, type II and on-off intermittencies are observed. Bifurcation diagram is plotted to facilitate the designer to choose a safe operating zone.